Variational Approach for a Robin Problem Involving Non Standard Growth Conditions

Abstract

In this manuscript, we study a Robin problem driven by the $p\left(x\right)$-Laplacian with two parameters. $-{\mathrm{div}\left(\right|\nabla w|}^{p\left(x\right)-2}{\nabla w)=\lambda V\left(x\right)|w|}^{q\left(x\right)-2}w,x\in \mathcal{Q}$, ${|\nabla w|}^{p\left(x\right)-2}\frac{\partial w}{\partial n}+{\theta \left(x\right)\left|w\right|}^{p\left(x\right)-2}=\beta V_1\left(x\right){\left|w\right|}^{r\left(x\right)-2}w.x\in \partial \mathcal{Q}.$ Here, $\mathcal{Q}$ is a regular bounded domain in ${\mathbb{R}}^N$, $\lambda ,\beta >0$, $p,q$ are continuous functions on $\overline{\mathcal{Q}}$, $\frac{\partial w}{\partial n}$ is the outer unit normal derivative on $\partial \mathcal{Q}$, $\theta \in L^{\infty }(\partial \mathcal{Q})$, such that ${\displaystyle ess\underset{x\in \partial \mathcal{Q}}{inf}\theta \left(x\right)>0,}$ V is an indefinite function in $L^{s\left(x\right)}\left(\mathcal{Q}\right)$ and $V_1$ is a non-negative one in $L^{s_1\left(x\right)}(\partial \mathcal{Q})$. Using variational tools, we show the existence of a non-trivial solution.

1. Introduction

This manuscript is concerned with a study of non-homogeneous Robin problem with the $p\left(x\right)-$Laplacian:

$$\left\{\begin{array}{cc}-{div\left(\right|\nabla w|}^{p\left(x\right)-2}{\nabla w)=\lambda V\left(x\right)|w|}^{q\left(x\right)-2}w,\hfill & x\in \mathcal{Q}\hfill \\ {|\nabla w|}^{p\left(x\right)-2}\frac{\partial w}{\partial n}+{\theta \left(x\right)\left|w\right|}^{p\left(x\right)-2}=\beta V_1\left(x\right){\left|w\right|}^{r\left(x\right)-2}w.\hfill & x\in \partial \mathcal{Q}\hfill \end{array}\right.$$

(1)

$\mathcal{Q}$ is a regular bounded domain in ${\mathbb{R}}^N$, $\lambda ,\beta >0$, the functions p and q are assumed to be continuous on $\overline{\mathcal{Q}}$, $\theta \in L^{\infty }(\partial \mathcal{Q})$ with ${\displaystyle ess\underset{x\in \partial \mathcal{Q}}{inf}\theta \left(x\right)>0,}$ V is a weight function in the generalised space $L^{s\left(x\right)}\left(\mathcal{Q}\right)$ and $V_1$ is a non-negative function in $L^{s_1\left(x\right)}(\partial \mathcal{Q})$.

The operator ${\Delta }_{p\left(x\right)}w:={div\left(\right|\nabla w|}^{p\left(x\right)-2}\nabla w)$ is a generalization of the classic p-Laplace operator, but due to the fact that ${\Delta }_{p\left(x\right)}$ is non-homogeneous, it has more complicated nonlinearity than the p-Laplace operator, so, we can find some difficulties in treating these kinds of operators; for example, the Lagrange Multiplier can not be used.

In recent decades, the study of $p\left(x\right)$-Laplace operators with Robin boundary conditions is very remarkable motivated by the modeling of thermo-convective flows of non-Newtonian fluids, as well as the electrorheological fluids (see [1]), elastic mechanics (see [2]), stationary thermo-rheological viscous flows of non-Newtonian fluids, image processing (see [3]), and the mathematical description of the processes filtration of a barotropic gas through a porous medium (see [4]). The reader can find recent contribution involving $p\left(x\right)$-Laplace operator in [5,6,7,8].

Moreover, Robin boundary conditions has been used in many physics problems, such that electromagnetic and heat transfer problems. In addition, Robin conditions are also needed to investigate Sturm–Liouville problems which are needed in different fields in sciences and engineering. The study on elliptic equation with Robin condition starts from the celebrated paper of S.G. Deng in [9]. The author is concerned with the problem

$$\left\{\begin{array}{cc}-\mathrm{div}\left(\right|\nabla w{|}^{p\left(x\right)-2}\nabla w)=\lambda f(x,w),\hfill & x\in \mathcal{Q}\hfill \\ {|\nabla w|}^{p\left(x\right)-2}\frac{\partial w}{\partial n}+\theta \left(x\right){\left|w\right|}^{p\left(x\right)-2}=0,\hfill & x\in \partial \mathcal{Q}\hfill \end{array}\right.$$

(2)

where $\mathcal{Q}$ is a regular bounded domain in ${\mathbb{R}}^N$, $\lambda >0$, p is a continuous function on $\overline{\mathcal{Q}}$, $a\in L^{\infty }(\partial \mathcal{Q}).$ Using the variational methods and under appropriate conditions, the author proved the existence of ${\lambda }_{*}>0$ under which problem (2) has at least two positive solutions, at least one positive solution for $\lambda ={\lambda }_{*}<+\infty$ and has no positive solution in the case when $\lambda >{\lambda }_{*}.$ After that many authors were devoted to the investigations for a variety of elliptic equations involving Robin boundary condition in a bounded domain. The readers can found a recent contributions in [6,10,11,12,13,14,15]. In [6], the author considered the Robin problem of type:

$$\left\{\begin{array}{cc}-{\mathrm{div}\left(\right|\nabla w|}^{p\left(x\right)-2}{\nabla w)=\lambda V\left(x\right)|w|}^{q\left(x\right)-2}w,\hfill & x\in \mathcal{Q}\hfill \\ {|\nabla w|}^{p\left(x\right)-2}\frac{\partial w}{\partial n}+\theta \left(x\right){\left|w\right|}^{p\left(x\right)-2}=0\hfill & x\in \partial \mathcal{Q},\hfill \end{array}\right.$$

(3)

where $\mathcal{Q}$ is a regular bounded domain in ${\mathbb{R}}^N$, $\lambda >0$, $p,q$ are continuous functions on $\overline{\mathcal{Q}}$, $a\in L^{\infty }(\partial \mathcal{Q})$ and V is an indefinite function in a generalised Lebesgue space $L^{s\left(x\right)}\left(\mathcal{Q}\right)$. The author proved several results related to the existence of a spectrum to problem (3). Moreover, the author obtained that any $\lambda >0$ is an eigenvalue for problem (3) using the variational method.

In this paper, we are interested to the existence of nontrivial solutions for problem (1). Here we follow the approach as in [6]. In fact, we use a variational approach to show that for any $\lambda ,\beta >0$, problem (1) has a nontrivial solution. We would like to remark that the results proved here are new even for the case that our class of indefinite weights used in this paper is larger than used in [9,10]. Moreover, this paper generalize the work of K. Kefi [6]. In fact, if we assume that $V_1\equiv 0$ in $\partial \mathcal{Q}$, problem (1) becomes problem (3) and we obtain the same result as in [6], so the result obtained in [6] appears as a particular case in our problem.

In what follows, the rest of the paper is articulated on two sections. First, we introduce some prerequisite properties of the generalized Lebesgue–Sobolev space $W^{1,p\left(x\right)}\left(\mathcal{Q}\right)$, second we state and prove the existence results concerning problem (1).

2. Backgrounds

In the following, we start by reminding properties and definitions for $L^{p\left(x\right)}\left(\mathcal{Q}\right)$ and $W^{1,p\left(x\right)}\left(\mathcal{Q}\right)$ (see [16]). Putting

$$C_{+}\left(\overline{\mathcal{Q}}\right):=\{f:f\in C\left(\overline{\mathcal{Q}}\right),f\left(x\right)>1,\mathrm{for} \mathrm{any}x\in \overline{\mathcal{Q}}\}$$

and for $p\in C_{+}\left(\overline{\mathcal{Q}}\right)$, let $p^{-}>1$, such that ${\displaystyle p^{-}:=\underset{x\in \overline{\mathcal{Q}}}{min}p\left(x\right)\le \underset{x\in \overline{\mathcal{Q}}}{max}p\left(x\right):=p^{+}<\infty }$ and

$$L^{p\left(x\right)}\left(\mathcal{Q}\right)=\{w:\mathcal{Q}\to \mathbb{R}\mathrm{measurable} \mathrm{with}{\int }_{\mathcal{Q}}{\left|w\left(x\right)\right|}^{p\left(x\right)}dx<\infty \}.$$

We equip this space using the Luxemburg norm

$${\left|w\right|}_{p\left(x\right)}=inf\{\gamma >0:{\int }_{\mathcal{Q}}|\frac{w\left(x\right)}{\gamma }{|}^{p\left(x\right)}dx\le 1\}.$$

We remark, when $p\left(x\right)=p$, $L^{p\left(x\right)}\left(\mathcal{Q}\right)$ becomes $L^p\left(\mathcal{Q}\right)$ with the standard norm ${\parallel w\parallel }_{L^p}=({\int }_{\mathcal{Q}}{\left|w\right|}^p{dx)}^{\frac{1}{p}}$.

In the sequel, define $L^{p^{{}^{\prime }}\left(x\right)}\left(\mathcal{Q}\right)$ as the conjugate of $L^{p\left(x\right)}\left(\mathcal{Q}\right)$, such that $\frac{1}{p}+\frac{1}{p^{{}^{\prime }}}=1$, we have the Hölder Inequality

$$|{\int }_{\mathcal{Q}}wvdx|\le (\frac{1}{p^{-}}+\frac{1}{{\left(p^{\prime }\right)}^{-}}){\left|w\right|}_{p\left(x\right)}{\left|v\right|}_{p^{\prime }\left(x\right)},w\in L^{p\left(x\right)}\left(\mathcal{Q}\right),v\in L^{p^{{}^{\prime }}\left(x\right)}\left(\mathcal{Q}\right).$$

(4)

Note, also, that if $f_1$, $f_2$, and $f_3:\overline{\mathcal{Q}}\to (1,\infty )$ are a Lipschitz continuous functions satisfying $1/f_1\left(x\right)+1/f_2\left(x\right)+1/f_3\left(x\right)=1$, then for any $u_1\in L^{f_1\left(x\right)}\left(\mathcal{Q}\right),u_2\in L^{f_2\left(x\right)}\left(\mathcal{Q}\right)$ and $u_3\in L^{f_3\left(x\right)}\left(\mathcal{Q}\right)$, we have (see [17], Proposition 2.5):

$$|{\int }_{\mathcal{Q}}{u}_1{u}_2{u}_{3}dx|\le \left(\frac{1}{f_1^{-}}+\frac{1}{{f_2}^{-}}+\frac{1}{{f_3}^{-}}\right)|u_1{|}_{f_1\left(x\right)}|u_2{|}_{f_2\left(x\right)}{|u_3|}_{f_3\left(x\right)}.$$

(5)

We define also the so called modular on $L^{p\left(x\right)}\left(\mathcal{Q}\right)$ which is a function ${\rho }_{p\left(x\right)}:L^{p\left(x\right)}\left(\mathcal{Q}\right)\to \mathbb{R}$ defined by

$${\rho }_{p\left(x\right)}\left(w\right):={\int }_{\mathcal{Q}}{|w|}^{p\left(x\right)}dx,$$

and verify

Proposition 1.

(See [18]) For any $w,v\in L^{p\left(x\right)}\left(\mathcal{Q}\right)$, one has

• ${|w|}_{p\left(x\right)}<1(resp.=1,>1)\iff {\rho }_{p\left(x\right)}\left(w\right)<1(resp.=1,>1).$
• ${min\left(\right|w|}_{p\left(x\right)}^{p^{-}}{,\left|w\right|}_{p\left(x\right)}^{p^{+}})\le {\rho }_{p\left(x\right)}\left(w\right)\le {max\left(\right|w|}_{p\left(x\right)}^{p^{-}}{,\left|w\right|}_{p\left(x\right)}^{p^{+}}).$
• ${\rho }_{p\left(x\right)}(w-v)\to 0\iff {|w-v|}_{p\left(x\right)}\to 0.$

Proposition 2.

(See [19]) Let p and q two measurable functions, such that $p\in L^{\infty }\left(\mathcal{Q}\right)$ and$1\le p\left(x\right)q\left(x\right)\le \infty$, for a.e. $x\in \mathcal{Q}$. Let $w\in L^{q\left(x\right)}\left(\mathcal{Q}\right)$, $w\ne 0$. Then,

$${min\left(\right|w|}_{p\left(x\right)q\left(x\right)}^{p^{+}}{,\left|w\right|}_{p\left(x\right)q\left(x\right)}^{p^{-}}{)\le ||w|}^{p\left(x\right)}{|}_{q\left(x\right)}\le {max\left(\right|w|}_{p\left(x\right)q\left(x\right)}^{p^{-}}{,\left|w\right|}_{p\left(x\right)q\left(x\right)}^{p^{+}}).$$

The reader can found more details for the modular in [16,18].

In the sequel, put $X:=W^{1,p\left(x\right)}\left(\mathcal{Q}\right)$, where

$$W^{1,p\left(x\right)}\left(\mathcal{Q}\right)=\{w\in L^{p\left(x\right)}\left(\mathcal{Q}\right):|\nabla w|\in L^{p\left(x\right)}\left(\mathcal{Q}\right)\}.$$

We equip this space with the norm

$${\displaystyle \parallel w\parallel =inf\{\eta >0:{\int }_{\mathcal{Q}}(|\frac{\nabla w}{\eta }{|}^{p\left(x\right)}dx+|\frac{w}{\eta }{|}^{p\left(x\right)})dx\le 1\}\mathrm{for}w\in X,}$$

$$\parallel w\parallel ={|\nabla w|}_{p\left(x\right)}+{\left|w\right|}_{p\left(x\right)}.$$

Then, $(X,\parallel .\parallel )$ is a Banach space which is separable and reflexive.

Putting

$${\displaystyle {\parallel w\parallel }_a=inf\{\eta >0:{\int }_{\mathcal{Q}}|\frac{\nabla w}{\eta }{|}^{p\left(x\right)}dx+{\int }_{\partial \mathcal{Q}}\theta \left(x\right)|\frac{w}{\eta }{|}^{p\left(x\right)}d{\sigma }_x\le 1\}\mathrm{for}w\in X.}$$

Then, due to Theorem 2.1 in [9], ${\parallel w\parallel }_a$ is a norm on X which is equivalent to $\parallel w\parallel$, moreover, if we define $I_a:X\to \mathbb{R}$ by

$$I_a\left(w\right)={\int }_{\mathcal{Q}}{|\nabla w|}^{p\left(x\right)}dx+{\int }_{\partial \mathcal{Q}}\theta \left(x\right){\left|w\right|}^{p\left(x\right)}d{\sigma }_x,$$

we have

Proposition 3.

(See [9])

• ${min(\parallel w\parallel }_a^{p^{-}}{,\parallel w\parallel }_a^{p^{+}})\le I_a\left(w\right)\le {max(\parallel w\parallel }_a^{p^{-}}{,\parallel w\parallel }_a^{p^{+}}).$
• $\parallel w_n{\parallel }_a\to 0(respectively\to \infty )\iff I_a\left(w_n\right)\to 0(respectively\to \infty ).$

Proposition 4.

(Proposition 2.2 in [20]) Let ${\displaystyle \xi \left(w\right)={\int }_{\mathcal{Q}}\frac{1}{p\left(x\right)}{|\nabla w|}^{p\left(x\right)}dx+{\int }_{\partial \mathcal{Q}}\frac{\theta \left(x\right)}{p\left(x\right)}{\left|w\right|}^{p\left(x\right)}d{\sigma }_x}$, then the function ${\xi }^{\prime }:X\to X^{*}$ is a strictly monotone, where ${\xi }^{\prime }$ is the derivative of ξ and $X^{*}$ is the dual space of X. Moreover, it is a continuous bounded homeomorphism and is of type $\left(S_{+}\right)$, so if $w_n\rightharpoonup w$ and ${\displaystyle \underset{n\to +\infty }{lim\; sup}{\xi }^{\prime }\left(w_n\right)(w_n-w)\le 0}$, then $w_n\to w$.

Now, let

$$p^{*}\left(y\right)=\left\{\begin{array}{c}{\displaystyle \frac{Np\left(y\right)}{N-p\left(y\right)},p\left(y\right)<N,}\hfill \\ {\displaystyle +\infty ,p\left(y\right)\ge N.}\hfill \end{array}\right.$$

We mention that if $l\in C^{+}\left(\overline{\mathcal{Q}}\right)$ and $l\left(y\right)<p^{*}\left(y\right)$ for all $x\in \overline{\mathcal{Q}}$, so $X\hookrightarrow L^{l\left(x\right)}\left(\mathcal{Q}\right)$ is compact.

For a measurable function $\theta :\partial \mathcal{Q}\to \mathbb{R}$, define the weighted space

$${\displaystyle L_{\theta \left(x\right)}^{p\left(x\right)}(\partial \mathcal{Q}):=\left\{w|w:\partial \mathcal{Q}\to \mathbb{R}\mathrm{is}\mathrm{a}\mathrm{measurable}{\int }_{\partial \mathcal{Q}}{\left|\theta \left(x\right)\right|\left|w\left(x\right)\right|}^{p\left(x\right)}d{\sigma }_x<+\infty \right\}}$$

we equip it with

$${\displaystyle {\left|w\right|}_{\left(p\left(x\right),\theta \left(x\right)\right)}={\left|w\right|}_{L_{\theta \left(x\right)}^{p\left(x\right)}(\partial \mathcal{Q})}=inf\left\{\tau >0:{\int }_{\partial \mathcal{Q}}\left|\theta \left(x\right)\right||\frac{w\left(x\right)}{\tau }{|}^{p\left(x\right)}d{\sigma }_x\le 1\right\},}$$

here $d{\sigma }_x$ is the measure on the boundary. Then, ${\displaystyle L_{\theta \left(x\right)}^{p\left(x\right)}(\partial \mathcal{Q})}$ is a Banach space. Note that in the particular case when $\theta \left(x\right)\equiv 1$ on ${\displaystyle \partial \mathcal{Q}}$, ${\displaystyle L_{\theta \left(x\right)}^{p\left(x\right)}(\partial \mathcal{Q})=L^{p\left(x\right)}(\partial \mathcal{Q})}$.

Proposition 5.

Let ${\displaystyle \rho \left(w\right)={\int }_{\partial \mathcal{Q}}{\left|\theta \left(x\right)\right|\left|w\left(x\right)\right|}^{p\left(x\right)}d{\sigma }_x}$. For $w,w_k\in L_{\theta \left(x\right)}^{p\left(x\right)}(\partial \mathcal{Q})(k=1,2,...)$, we have

(1) 

${\displaystyle {\left|w\right|}_{\left(p\left(x\right),\theta \left(x\right)\right)}\ge 1\iff {\left|w\right|}_{\left(p\left(x\right),\theta \left(x\right)\right)}^{p^{-}}\le \rho \left(w\right)\le {\left|w\right|}_{\left(p\left(x\right),\theta \left(x\right)\right)}^{p^{+}};}$

(2) 

${\displaystyle {\left|w\right|}_{\left(p\left(x\right),\theta \left(x\right)\right)}\le 1\iff {\left|w\right|}_{\left(p\left(x\right),\theta \left(x\right)\right)}^{p^{+}}\le \rho \left(w\right)\le {\left|w\right|}_{\left(p\left(x\right),\theta \left(x\right)\right)}^{p^{-}};}$

(3) 

${\displaystyle |w_k{|}_{\left(p\left(x\right),\theta \left(x\right)\right)}\to 0\iff \rho \left(w_k\right)\to 0}$;

(4) 

${\displaystyle |w_k{|}_{\left(p\left(x\right),\theta \left(x\right)\right)}\to \infty \iff \rho \left(w_k\right)\to \infty }$.

For $K\subset \overline{\mathcal{Q}}$, we denote by ${\displaystyle p^{-}\left(K\right)=\underset{y\in K}{inf}p\left(x\right)}$ and ${\displaystyle p^{+}\left(K\right):=\underset{y\in K}{sup}p\left(y\right)}$. Define

$$p^{\partial }\left(y\right)={\left(p\left(y\right)\right)}^{\partial }=\left\{\begin{array}{c}{\displaystyle \frac{(N-1)p\left(y\right)}{N-p\left(y\right)},\mathrm{if}p\left(y\right)<N,}\hfill \\ \infty ,\mathrm{if}p\left(y\right)\ge N,\hfill \end{array}\right.$$

$${\displaystyle p_{\gamma \left(y\right)}^{\partial }\left(y\right):=\frac{\gamma \left(y\right)-1}{\gamma \left(y\right)}{p}^{\partial }\left(y\right),}$$

where $x\in \partial \mathcal{Q},\gamma \in C(\partial \mathcal{Q})$ with ${\displaystyle {\gamma }^{-}=\underset{x\in \partial \mathcal{Q}}{inf}\gamma \left(x\right)>1}$.

Our conditions are the following:

Hypothesis 1.

${\displaystyle 1<q\left(x\right)<p\left(x\right)<N<s\left(x\right)for allx\in \overline{\mathcal{Q}}}$, $V\in L^{s\left(x\right)}\left(\mathcal{Q}\right)$ and $V>0$ in ${\mathcal{Q}}_0\subset \subset \mathcal{Q}$, with $|{\mathcal{Q}}_0|>0$.

Hypothesis 2.

${\displaystyle 1\le max\{r^{+},q^{+}\}<p^{-},p^{-}(\partial \mathcal{Q})<p^{\partial }\left(x\right),\frac{N-1}{p\left(x\right)-1}<s_1\left(x\right)}$ for all $x\in \partial \mathcal{Q}$ and $V_1$ is a non-negative function in $L^{s_1\left(x\right)}(\partial \mathcal{Q}).$

We recall the next important theorem.

Theorem 1.

(See Theorem 2.1 [9]) Suppose that the cone property holds on the boundary of $\mathcal{Q}$ and $p\in C\left(\overline{\mathcal{Q}}\right)$, such that $1<p^{-}$. Assume, also, that $\theta \in L^{\gamma \left(x\right)}(\partial \mathcal{Q}),\gamma \in C(\partial \mathcal{Q})$ and ${\displaystyle \gamma \left(x\right)>\frac{p^{\partial }\left(x\right)}{p^{\partial }\left(x\right)-1}}$ for all $x\in \partial \mathcal{Q}$. If $q\in C(\partial \mathcal{Q})$ and

$${\displaystyle 1\le q\left(x\right)<p_{\gamma \left(x\right)}^{\partial }\left(x\right),for allx\in \partial \mathcal{Q}.}$$

(6)

one has, $W^{1,p\left(x\right)}\left(\mathcal{Q}\right)\hookrightarrow L_{\theta \left(x\right)}^{q\left(x\right)}(\partial \mathcal{Q})$ is compact. Consequently, ${\displaystyle W^{1,p\left(x\right)}\left(\mathcal{Q}\right)\hookrightarrow L^{q_0\left(x\right)}(\partial \mathcal{Q})}$ is compact where $1\le q_0\left(x\right)<p^{\partial }\left(x\right)$ for all $x\in \partial \mathcal{Q}$.

Definition 1.

We said that $w\in X\backslash \left\{0\right\}$ is a weak solution of problem (1) if

$${\int }_{\mathcal{Q}}{|\nabla w|}^{p\left(x\right)-2}\nabla w\nabla vdx+{\int }_{\partial \mathcal{Q}}{\theta \left(x\right)\left|w\right|}^{p\left(x\right)-2}wvd\sigma -\lambda {\int }_{\mathcal{Q}}{V\left(x\right)\left|w\right|}^{q\left(x\right)-2}wvdx-\beta {\int }_{\partial \mathcal{Q}}{V}_1\left(x\right){\left|w\right|}^{r\left(x\right)-2}wvd\sigma =0,$$

for any $v\in X$.

3. Results

In what follows, denote by $c,c_i,i=1,2,...$ a positive constants which change from line to another.

Put $s^{{}^{\prime }}\left(x\right)$, $s_1^{{}^{\prime }}\left(x\right)$ the conjugate functions of $s\left(x\right)$, $s_1\left(x\right)$, respectively, and let

$$\alpha \left(x\right):=\frac{s\left(x\right)q\left(x\right)}{s\left(x\right)-q\left(x\right)}\mathrm{and}{\alpha }_1\left(x\right):=\frac{s_1\left(x\right)r\left(x\right)}{s_1\left(x\right)-r\left(x\right)},$$

then we have:

Remark 1.

Assume that conditions $\left({\mathbf{H}}_{\mathbf{1}}\right)$ and $\left({\mathbf{H}}_{\mathbf{2}}\right)$ hold, then

$$s{}^{\prime }\left(x\right)q\left(x\right)<p^{*}\left(x\right),for allx\in \overline{\mathcal{Q}},$$

resp. $s{{}^{\prime }}_1\left(x\right)r\left(x\right)<p^{\partial }\left(x\right)$ for all $x\in \partial \mathcal{Q}$,

$$\alpha \left(x\right)<p^{*}\left(x\right),for allx\in \overline{\mathcal{Q}},$$

resp. ${\alpha }_1\left(x\right)<p^{\partial }\left(x\right)$ for all $x\in \partial \mathcal{Q}.$ Consequently, the embedding

$$X\hookrightarrow L^{s^{{}^{\prime }}\left(x\right)q\left(x\right)}\left(\mathcal{Q}\right)andX\hookrightarrow L^{\alpha \left(x\right)}\left(\mathcal{Q}\right),$$

resp. $X\hookrightarrow L^{s_1^{{}^{\prime }}\left(x\right)r\left(x\right)}(\partial \mathcal{Q})andX\hookrightarrow L^{{\alpha }_1\left(x\right)}(\partial \mathcal{Q}),$ are compact and continuous.

Our result is due to Theorem 25.D in [21].

Theorem 2.

Suppose that conditions $\left({\mathbf{H}}_{\mathbf{1}}\right)$and$\left({\mathbf{H}}_{\mathbf{2}}\right)$ hold. Then, for any $\lambda ,\beta >0$ problem (1) has a weak solution.

In the following, let the functions $\mathsf{\Phi }$, $J_1\mathrm{and} J_2:X\to \mathbb{R}$ defined by:

$${\displaystyle \mathsf{\Phi }\left(w\right)={\int }_{\mathcal{Q}}\frac{1}{p\left(x\right)}{|\nabla w|}^{p\left(x\right)}dx+{\int }_{\partial \mathcal{Q}}\frac{\theta \left(x\right)}{p\left(x\right)}{\left|w\right|}^{p\left(x\right)}d\sigma ,J_1\left(w\right)={\int }_{\mathcal{Q}}\frac{V\left(x\right)}{q\left(x\right)}{\left|w\right|}^{q\left(x\right)}dx\mathrm{and} J_2\left(w\right)={\int }_{\partial \mathcal{Q}}\frac{V_1\left(x\right)}{r\left(x\right)}{\left|w\right|}^{r\left(x\right)}d\sigma .}$$

Using Proposition 2 and Remark 1, we assert that $J_1$ and $J_2$ are well-defined. In fact, for all $w\in X$, one has

$$\begin{array}{c}\hfill \begin{array}{cc}{\displaystyle |J_1\left(w\right)|}\hfill & \le \frac{1}{q^{-}}{\left|V\right|}_{s\left(x\right)}{||w|}^{q\left(x\right)}{|}_{s^{{}^{\prime }}\left(x\right)}\le \frac{1}{q^{-}}{\left|V\right|}_{s\left(x\right)}{max\left(\right|w|}_{s^{{}^{\prime }}\left(x\right)q\left(x\right)}^{q^{-}}{,\left|w\right|}_{s^{{}^{\prime }}\left(x\right)q\left(x\right)}^{q^{+}}),\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{cc}{\displaystyle |J_2\left(w\right)|}\hfill & \le \frac{1}{r^{-}}|V_1{|}_{s_1\left(x\right)}{||w|}^{r\left(x\right)}{|}_{s_1^{\prime }\left(x\right)}\le \frac{1}{r^{-}}|V_1{|}_{s_1\left(x\right)}{max\left(\right|w|}_{s_1^{\prime }\left(x\right)r\left(x\right)}^{r^{-}}{,|w|}_{s_1^{\prime }\left(x\right)r\left(x\right)}^{r^{+}}).\hfill \end{array}\end{array}$$

The energy of the problem (1) is ${\mathsf{\Psi }}_{\lambda ,\beta }:X\to \mathbb{R}$, where

$${\mathsf{\Psi }}_{\lambda ,\beta }\left(w\right):=\mathsf{\Phi }\left(w\right)-\lambda J_1\left(w\right)-\beta J_2\left(w\right).$$

First, we show some auxiliary results:

Proposition 6.

Assume that the assertions $({\mathbf{H}}_{\mathbf{1}})$ and $({\mathbf{H}}_{\mathbf{2}})$ are fulfilled, then ${\mathsf{\Psi }}_{\lambda ,\beta }$ is weakly lower semi-continuous and inside the space $C^1(X,\mathbb{R})$, moreover $w\in X$ is a critical point of ${\mathsf{\Psi }}_{\lambda ,\beta }$ if, and only if, w is a weak solution for problem (1).

For the proof of Proposition 6, we advice readers to refer to K. Kefi [6].

Now, we show the existence of a valley for the functional ${\mathsf{\Psi }}_{\lambda ,\beta }$ near the origin

Lemma 1.

There exists ${\phi }_0\in X$, such that ${\phi }_0\ge 0$, ${\phi }_0\ne 0$, and ${\mathsf{\Psi }}_{\lambda ,\beta }\left(t{\phi }_0\right)<0$, for $t>0$ small enough.

Proof. 

Due to assumption $\left({\mathbf{H}}_{\mathbf{1}}\right)$, we have $q\left(x\right)<p\left(x\right)\mathrm{for}\mathrm{all}x\in {\mathcal{Q}}_0.$ In the following, let $q_0^{-}=in{f}_{{\mathcal{Q}}_0}q\left(x\right)$ and $p_0^{-}=in{f}_{{\mathcal{Q}}_0}p\left(x\right).$ Let ${ϵ}_0$ such that $q_0^{-}+{ϵ}_0<p_0^{-}.$ Since $q\in C\left({\overline{\mathcal{Q}}}_0\right)$, then there exists an open set ${\mathcal{Q}}_1\subset {\mathcal{Q}}_0$ verifying $|q\left(x\right)-q_0^{-}|<{ϵ}_0\mathrm{for}\mathrm{all}x\in {\mathcal{Q}}_1.$ So $q\left(x\right)\le q_0^{-}+{ϵ}_0<p_0^{-},\mathrm{for}\mathrm{all}x\in {\mathcal{Q}}_1.$

Now, let ${\phi }_0\in C_0^{\infty }\left(\mathcal{Q}\right)$, such that $supp\left({\phi }_0\right)\subset {\mathcal{Q}}_1\subset {\mathcal{Q}}_0$, ${\phi }_0=1$ in a subset ${\mathcal{Q}}_1^{{}^{\prime }}\subset supp\left({\phi }_0\right)$, $0\le {\phi }_0\le 1$ in ${\mathcal{Q}}_1$. Then,

$$\begin{array}{ccc}\hfill {\mathsf{\Psi }}_{\lambda ,\beta }\left(t{\phi }_0\right)& =& {\int }_{\mathcal{Q}}\frac{1}{p\left(x\right)}\left(|\nabla \left(t{\phi }_0\right){|}^{p\left(x\right)}+\theta \left(x\right){|t{\phi }_0|}^{p\left(x\right)}\right)dx-\beta {\int }_{\partial \mathcal{Q}}{V}_1\left(x\right)|t{\phi }_0{|}^{r\left(x\right)}d{\sigma }_x-\lambda {\int }_{\mathcal{Q}}V\left(x\right){|t{\phi }_0|}^{q\left(x\right)}dx\hfill \\ & \le & \frac{1}{p_0^{-}}{\int }_{{\mathcal{Q}}_0}{t}^{p\left(x\right)}\left(|\nabla {\phi }_0{|}^{p\left(x\right)}+\theta \left(x\right){|{\phi }_0|}^{p\left(x\right)}\right)dx-\lambda {\int }_{{\mathcal{Q}}_1}{t}^{q\left(x\right)}V\left(x\right){|{\phi }_0|}^{q\left(x\right)}dx\hfill \\ & \le & \frac{t^{p_0^{-}}}{p_0^{-}}{I}_a\left({\phi }_0\right)-\lambda t^{q_0^{-}+{ϵ}_0}{\int }_{{\mathcal{Q}}_1}V\left(x\right){|{\phi }_0|}^{q\left(x\right)}dx\hfill \\ & \le & \frac{t^{p_0^{-}}}{p_0^{-}}max(\parallel {\phi }_0{\parallel }_a^{p^{+}},\parallel {\phi }_0{\parallel }_a^{p^{-}})-\lambda t^{q_0^{-}+{ϵ}_0}{\int }_{{\mathcal{Q}}_1}V\left(x\right){|{\phi }_0|}^{q\left(x\right)}dx,\hfill \end{array}$$

consequently

$${\displaystyle {\mathsf{\Psi }}_{\lambda ,\beta }\left(t{\phi }_0\right)<0,}$$

for ${\displaystyle t<{\delta }^{1/(p_0^{-}-q_0^{-}-{ϵ}_0)}}$ with

$${\displaystyle 0<\delta <min\left\{1,\frac{\lambda p_0^{-}{\int }_{{\mathcal{Q}}_1}V\left(x\right){\left|{\phi }_0\right|}^{q\left(x\right)}dx}{max(\parallel {\phi }_0{\parallel }_a^{p^{+}},\parallel {\phi }_0{\parallel }_a^{p^{-}})}\right\}.}$$

Since ${\phi }_0=1$ in ${\mathcal{Q}}_1^{{}^{\prime }}$, then $\parallel {\phi }_0{\parallel }_a>0$, which completes the proof of Lemma 1. □

Lemma 2.

Assume the conditions $\left({\mathbf{H}}_{\mathbf{1}}\right)$ and $\left({\mathbf{H}}_{\mathbf{2}}\right)$ are fulfilled, then the functional ${\mathsf{\Psi }}_{\lambda ,\beta }$ is coercive.

Proof. 

Let $w\in X$ with ${\parallel w\parallel }_a>1$, so, from the Hölder inequality (4) and the Sobolev embedding, one has

$$\begin{array}{ccc}\hfill {\mathsf{\Psi }}_{\lambda ,\beta }\left(w\right)& =& {\int }_{\mathcal{Q}}\frac{1}{p\left(x\right)}{|\nabla w|}^{p\left(x\right)}dx+{\int }_{\partial \mathcal{Q}}\theta \left(x\right){\left|w\right|}^{p\left(x\right)}d{\sigma }_x-\frac{\lambda }{q^{-}}{\int }_{\mathcal{Q}}V\left(x\right){\left|w\right|}^{q\left(x\right)}dx-\frac{\beta }{r^{-}}{\int }_{\partial \mathcal{Q}}{V}_1\left(x\right){\left|w\right|}^{r\left(x\right)}dx\hfill \\ & \ge & \frac{1}{p^{+}}{\parallel w\parallel }_a^{p^{-}}-\frac{\beta }{r^{-}}{C}_2|V_1{|}_{s_1\left(x\right)}{\left|w\right|}_{L^{s_1^{{}^{\prime }}\left(x\right)r\left(x\right)}(\partial \mathcal{Q})}^{r^{-}}-\frac{\lambda }{q^{-}}{C}_1{\left|V\right|}_{s\left(x\right)}{\left|w\right|}_{L^{s^{{}^{\prime }}\left(x\right)q\left(x\right)}\left(\mathcal{Q}\right)}^{q^{-}}\hfill \\ & \ge & \frac{1}{p^{+}}{\parallel w\parallel }_a^{p^{-}}-\beta C_2|V_1{|}_{s_1\left(x\right)}{C}_4^{r^{-}}{\parallel w\parallel }_a^{r^{-}}-\lambda C_1{\left|V\right|}_{s\left(x\right)}{C}_3^{q^{-}}{\parallel w\parallel }_a^{q^{-}}\hfill \\ & \ge & \frac{1}{p^{+}}{\parallel w\parallel }_a^{p^{-}}-\lambda C_1{\left|V\right|}_{s\left(x\right)}{C}_3^{q^{-}}{\parallel w\parallel }_a^{q^{-}}-\beta C_2|V_1{|}_{s_1\left(x\right)}{C}_4^{r^{-}}{\parallel w\parallel }_a^{r^{-}}.\hfill \end{array}$$

Since $max(q^{-},r^{-})<p^{-}$, then ${\displaystyle \underset{w\to +\infty }{lim}{\mathsf{\Psi }}_{\lambda ,\beta }\left(w\right)=0}$, so ${\mathsf{\Psi }}_{\lambda ,\beta }$ is coercive. □

Proof of Theorem 2 completed.

${\mathsf{\Psi }}_{\lambda ,\beta }$ has a global minimizer. In addition, due to Lemma 1 this minimizer is non-trivial which complete the proof of our result. □